In this lesson I will show how to numerically solve algebraic and ordinary differential equations, and perform numerical integration with Simpson method. I will start with the solution of algebraic equations. The secant method is one of the simplest methods for solving algebraic equations. It is usually used as a part of a larger algorithm to improve convergence. As in any numerical algorithm, we need to check that the method is converging to a given precision in a certain number of steps. This is a precaution to avoid an infinite loop.
//secant method
using System;
class Secant
{
public delegate double Function(double x);
//declare a delegate that takes double and returns double
public static void secant(int step_number, double point1,double point2,Function f)
{
double p2,p1,p0,prec=.0001f; //set
precision to .0001
int i;
p0=f(point1);
p1=f(point2);
p2=p1-f(p1)*(p1-p0)/(f(p1)-f(p0)); //secant
formula
for(i=0;System.Math.Abs(p2)>prec &&i<step_number;i++)
//iterate till precision goal is not met or the maximum
//number of steps is reached
{
p0=p1;
p1=p2;
p2=p1-f(p1)*(p1-p0)/(f(p1)-f(p0));
}
if(i<step_number)
Console.WriteLine(p2);
//method converges
else
Console.WriteLine("{0}.The method did not converge",p2);//method
does not converge
}
}
class Demo
{//equation f1(x)==0;
public static double f1( double x)
{
return x*x*x-2*x-5;
}
public static void Main()
{
Secant.secant(5,0,1,new Secant.Function(f1));
}
}
Our second example is a Simpson integration algorithm. We have introduced numerical integration in our discussion of delegates in Lesson12. The Simpson algorithm is more precise than the naive integration algorithm I have used there. The basic idea of the Simpson algorithm is to sample the integrand in a number of points to get a better estimate of its variations in a given interval. So Simpson method is more precise than the method shown in Lesson12, however since Simpson method samples more points it is slower.
//Simpson integration algorithm
using System;
//calculate the integral of f(x) between x=a and x=b by spliting the interval in step_number steps
class Integral
{
public delegate double Function(double x); //declare a delegate that takes
and returns double
public static double integral(Function f,double a, double b,int
step_number)
{
double sum=0;
double step_size=(b-a)/step_number;
for(int i=0;i<step_number;i=i+2) //Simpson
algorithm samples the integrand in several point which significantly improves
//precision.
sum=sum+(f(a+i*step_size)+4*f(a+(i+1)*step_size)+f(a+(i+2)*step_size))*step_size/3;
//divide the area under f(x) //into
step_number rectangles and sum their areas
return sum;
}
}
class Test
{
//simple functions to be integrated
public static double f1( double x)
{
return x*x;
}
public static double f2(double x)
{
return x*x*x;
}
public static void Main()
{//output the value of the integral.
Console.WriteLine(Integral.integral(new Integral.Function(f1),1,10,20));
}
}
Finally, let me show a simple code for solving first order ordinary differential equations. The code uses a Runge-Kutta method. The simplest method to solve ODE is to do a Taylor expansion, which is called Euler's method. Euler's method approximates the solution with the series of consecutive secants. The error in Euler's method is O(h) on every step of size h. The Runge-Kutta method has an error O(h^4)
using System;
//fourth order Runge Kutte method for y'=f(t,y);
//solve first order ode in the interval (a,b) with a given initial condition at x=a and fixed step h.
class Runge{
public delegate double Function(double t,double y); //declare a delegate that takes a double and returns
//double
public static void runge(double a, double b,double value, double step, Function f)
{
double t,w,k1,k2,k3,k4;
t=a;
w=value;
for(int i=0;i<(b-a)/step;i++){
k1=step*f(t,w);
k2=step*f(t+step/2,w+k1/2);
k3=step*f(t+step/2,w+k2/2);
k4=step*f(t+step,w+k3);
w=w+(k1+2*k2+2*k3+k4)/6;
t=a+i*step;
Console.WriteLine("{0} {1} ",t,w);
}
}
}
class Test
{
public static double f1(double t, double y)
{
return -y+t+1;
}
public static void Main()
{
Runge.runge(0,1,1,.1f,new Runge.Function(Test.f1));
}
}
Runge-Kutta methods with a variable step size are often used in
practice since they converge faster than fixed size methods.